Large-scale energy and pressure dynamics in decaying 2D incompressible isotropic turbulence

Abstract

Numerical simulations using either molecular or hyper-viscosity are carried out to study the temporal evolution of large-scale energy and pressure statistics in decaying two-dimensional incompressible isotropic turbulence. Initial Gaussian velocity fields peaking at small scales are considered. For each set of initial parameters, multiple realizations are performed to achieve reasonable statistical convergence. A wide range of Reynolds numbers (based on an equivalent Taylor microscale) is explored. For an initial energy spectrum E(k,0) ∝ k s 0 as k → 0 with s 0 ≥ 3, and at high enough Reynolds number, the numerical simulations display an important energy backscatter at subsequent times: E(k,t) ∝ t γ e k s , where s ≈ 3 and γ e converges at high times towards 2.5. The pressure spectrum E pp (k,0) is initially proportional to k 1 at small wavenumbers, in agreement with the predictions of quasi-normal theory. After a short transient decay, the infrared pressure spectrum increases significantly with time, while becoming steeper than k 1. However, a Gaussian randomization of the velocity allows the k 1 pressure spectrum to be recovered. In the high-Reynolds-number regime, the infrared pressure spectrum increases enough to induce a temporal growth of the pressure variance. We examine self-similar behaviours based on Taylor, integral and dissipative scales. Finally, we determine pressure pdf's, which compare favourably with earlier analytic predictions based on shell models made by Holzer and Siggia. LEGI is a joint laboratory of the Centre National de la Recherche Scientifique (CNRS), Université Joseph Fourier and the Institut National Polytechnique de Grenoble.